The lifespans of porcupines in a particular zoo are normally distributed. The average porcupine lives $18.2$ years; the standard deviation is $2.4$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a porcupine living between $23$ and $25.4$ years.
Solution: $18.2$ $15.8$ $20.6$ $13.4$ $23$ $11$ $25.4$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $18.2$ years. We know the standard deviation is $2.4$ years, so one standard deviation below the mean is $15.8$ years and one standard deviation above the mean is $20.6$ years. Two standard deviations below the mean is $13.4$ years and two standard deviations above the mean is $23$ years. Three standard deviations below the mean is $11$ years and three standard deviations above the mean is $25.4$ years. We are interested in the probability of a porcupine living between $23$ and $25.4$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the porcupines will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the porcupines will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of porcupines between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular porcupine living between $23$ and $25.4$ years is $\color{orange}{2.35\%}$.